We describe how certain cyclotomic Nazarov-Wenzl algebras occur as endomorphism rings of projective modules in a parabolic version of BGG category O of type D. Furthermore we study a family of subalgebras of these endomorphism rings which exhibit similar behaviour to the family of Brauer algebras even when they are not semisimple. The translation functors on this parabolic category O are studied and proven to yield a categorification of a coideal subalgebra of the general linear Lie algebra. Finally this is put into the context of categorifying skew Howe duality for these subalgebras.

For each integer k ≥ 4, we describe diagrammatically a positively graded Koszul algebra D k such that the category of finite dimensional D k -modules is equivalent to the category of perverse sheaves on the isotropic Grassmannian of type D k or B k−1 , constructible with respect to the Schubert stratification. The algebra is obtained by a (non-trivial) "folding" procedure from a generalized Khovanov arc algebra. Properties such as graded cellularity and explicit closed formulas for graded decomposition numbers are established by elementary tools.Mathematics Subject Classification 05E10 · 14M15 · 17B10 · 17B45 · 55N91 · 20C08

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Abstract. Blanchet introduced certain singular cobordisms to fix the functoriality of Khovanov homology. In this paper we introduce graded algebras consisting of such singular cobordismsà la Blanchet. As the main result we explicitly describe these algebras in algebraic terms using the combinatorics of arc diagrams.

We prove that the bigraded, colored Khovanov–Rozansky type A link and tangle invariants are functorial with respect to link and tangle cobordisms.